L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 27.7·5-s + 216·6-s + 679.·7-s + 512·8-s + 729·9-s − 221.·10-s − 6.56e3·11-s + 1.72e3·12-s − 651.·13-s + 5.43e3·14-s − 749.·15-s + 4.09e3·16-s − 2.72e4·17-s + 5.83e3·18-s + 2.33e4·19-s − 1.77e3·20-s + 1.83e4·21-s − 5.25e4·22-s − 7.96e4·23-s + 1.38e4·24-s − 7.73e4·25-s − 5.21e3·26-s + 1.96e4·27-s + 4.34e4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0992·5-s + 0.408·6-s + 0.748·7-s + 0.353·8-s + 0.333·9-s − 0.0701·10-s − 1.48·11-s + 0.288·12-s − 0.0822·13-s + 0.529·14-s − 0.0573·15-s + 0.250·16-s − 1.34·17-s + 0.235·18-s + 0.779·19-s − 0.0496·20-s + 0.432·21-s − 1.05·22-s − 1.36·23-s + 0.204·24-s − 0.990·25-s − 0.0581·26-s + 0.192·27-s + 0.374·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 59 | \( 1 - 2.05e5T \) |
good | 5 | \( 1 + 27.7T + 7.81e4T^{2} \) |
| 7 | \( 1 - 679.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.56e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 651.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.72e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.33e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.96e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.73e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.34e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.32e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.48e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.83e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.82e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.15e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.08e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.22e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.97e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.06e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.79e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.23e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.991941861857923957889502843506, −8.653113940810893571945614818113, −7.87962342636026772485859778197, −7.07206150250081429803911959750, −5.69345542021702142785073569655, −4.82668252902009624516507475072, −3.82864809152726333902754820748, −2.58855226132018344745724984918, −1.78977867332551141396450476822, 0,
1.78977867332551141396450476822, 2.58855226132018344745724984918, 3.82864809152726333902754820748, 4.82668252902009624516507475072, 5.69345542021702142785073569655, 7.07206150250081429803911959750, 7.87962342636026772485859778197, 8.653113940810893571945614818113, 9.991941861857923957889502843506